Page 34 - DMTH403_ABSTRACT_ALGEBRA

Unit 2: Groups

Notes
Example: Let G be the set of all 2 x 2 matrices with non-zero determinant. That is,

  a b  
0

G   c d  a, b, c, d r, ad – bc  
    
Consider G with the usual matrix multiplication, i.e, for
 a b  p q  ap br aq bs


A    and p    in G, A.P   


 c d   r s   cp dr cq ds 
Show that (G, ‘) is a group.
Solution: First we show that . is a binary operation, that is, A, P  G A.P  G.
Now,
det (A.P) = det A. det P # 0. since det A  0, det P  0.
Hence, A.P  G for all A, P in G.

Note det (AB) = (det A) (det B)

We also know that matrix multiplication is associative and   1 0  is the multiplicative identity.
 0 1 

Now, for A =   a b  in G. the mamx
 c d 

 d  b 
 ad bc ad bc  *  1 0


B    is such that det B   0 and AB    .

  c a  ad bc  0 1 
  ad bc ad bc  


Thus, B = A . (Note that we have used the axiom G3' here, and not G3.) This shows that the set of
–1
all 2 × 2 matrices over R with non-zero determinant forms a group under multiplication. Since
 1 2  0 1  2 1
 
 3 4 1 0    4 3  and
     
 0 1  1 2  3 4
 1 0   3 4    1 2 
     
we see that this group is not commutative.
And now another example of an abelian group.

Example: Consider the set of all translations of R ,
2
2
2


T  {f a,b : R  R |f (x, y) (x a, y b) for some fixed a, b R}


a,b
Note that each element f in T is represented by a point (a, b) in R . Show that (T, o) is a group,
2
a,b
where o denotes the composition of functions.
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